On Gradings Modulo 2 of Simple Lie Algebras in Characteristic 2

Symmetry Integrability and Geometry-Methods and Applications

article

Andrey Krutov¹  Alexei Lebedev² 

[1] Institute of Mathematics, Polish Academy of Sciences;Equa Simulation AB

关键词: Higgs bundle;    Morse–Bott theory;    localization;    Betti number;    moduli space;    stability;    quiver;    partition problem;   

DOI  :  10.3842/SIGMA.2018.130

来源: National Academy of Science of Ukraine

【 摘 要 】

The ground field in the text is of characteristic 2. The classification of modulo 2 gradings of simple Lie algebras is vital for the classification of simple finite-dimensional Lie superalgebras: with each grading, a simple Lie superalgebra is associated, see arXiv:1407.1695. No classification of gradings was known for any type of simple Lie algebras, bar restricted Jacobson-Witt algebras (i.e., the first derived of the Lie algebras of vector fields with truncated polynomials as coefficients) on not less than 3 indeterminates. Here we completely describe gradings modulo 2 for several series of Lie algebras and their simple relatives: of special linear series, its projectivizations, and projectivizations of the derived Lie algebras of two inequivalent orthogonal series (except for ${\mathfrak{o}}_\Pi(8)$). The classification of gradings is new, but all of the corresponding superizations are known. For the simple derived Zassenhaus algebras of height $n>1$, there is an $(n-2)$-parametric family of modulo 2 gradings; all but one of the corresponding simple Lie superalgebras are new. Our classification also proves non-triviality of a deformation of a simple $3|2$-dimensional Lie superalgebra (new result).

【 授权许可】

Unknown